First off, what type of Video Card (dual monitor card) did you purchase?
1.
PCI (Peripheral Component Interconnect)
2.
AGP (Accelerated/Advanced Graphics Port)
3.
PCI Express (PCIe) <--- this type of card is VERY different from the above listed PCI card. Unfortunately the developers of the standard didn't think that having a similar name to the old PCI would be that confusing, but I beg to differ.
Secondly, what type of expansion slot is available on your computer's motherboard? Most state-of-the-art video cards and motherboards available today are using the PCIe interface. Older computer motherboards and video cards generally used AGP (1997 - today). Before that, video cards were usually using the
VESA Local Bus (now obsolete) until PCI eventually took over in 1994-6 (1992 - today). Now, what does this mean for you?
Check to see where your monitor is connected currently. If the VGA port is located in the keyboard/mouse/USB connector blocks, then you have an integrated video graphics chipset. Now the question is, do you have a PCIe or AGP slot to upgrade your existing graphics? Some older computers that have integrated video (not using a discrete video card) did use an optional, empty AGP slot for upgrading/replacing the built-in video graphics, but there are quite a few that did not. In those cases, the only solution would be to use a PCI video card. Most newer integrated motherboards have a PCIe slot for graphics upgrades.
If the monitor is connected to a card, you are in luck! You just have to find out what type of card it is now (most likely AGP), and replace it with a dual monitor capable one.
As for adapters, well, technically it is possible to produce some slot/card combinations and there are some that are actually available. But there is the issue of card height (card + adapter = too tall to fit into the computer case unless you specifically get a low height video card). And the compromises involved in adapting different protocols to each other will definately mean reduced performance.